(A)symmetry

Antigoni Kaliontzopoulou, CIBIO/InBIO, University of Porto

12 October, 2019

Symmetry and Asymmetry

Types of Bilateral Symmetry

The Problem with Symmetrical Objects

The Problem with Symmetrical Objects

Symmetric Objects: Example

The Biology of Asymmetry

Types of Symmetry

Analysis of Symmetry: General Procedure

Extensions to GM Shape Data

Klingenberg & McIntyre (1998). Evolution.; Klingenberg et al. (2002). Evolution.

Decomposition of Asymmetry Component

\[\small{SST}=\sum^n_1{D}^2_{(X_i,Y_i)}=nD^2_{(\overline{X},\overline{Y})}+\sum^n_1{D}^2_{(X_i-\overline{X},Y_i-\overline{Y})}\]

Mardia et al. (2000). Biometrika.

Matching Symmetry: Example

Object Symmetry: Example

Other Types of Asymmetry

Symmetry Groups

Savriama and Klingenberg. (2011). BMC Evol. Biol.

Symmetry: Extended Protocol

Symmetry: Extended Protocol

Savriama and Klingenberg. (2011). BMC Evol. Biol.

Complex Symmetry: Example

A Note on Geometric Transformations

Geometric Transformations: Articulations

For articulated structures, several solutions exist - Fixing the angle in all specimens through a mathematical transformation - Separating the subsets to analyse separately, etc.

Adams. (1999). Evol. Ecol. Res.

Articulation Standardization: Example

Another Note on Geometric Transformations

NOTE: see Phenotypic Trajectory Analysis lecture for analytical details

Symmetry: Summary